On the double of the (restricted) super Jordan plane

Abstract: We consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in works of N. Andruskiewitsch, I. Angiono, I. Heckenberger, and its restricted version in odd characteristic introduced by the same authors. We show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in their work. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of $\mathfrak{osp}(1|2)$.